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APPEARED IN BULLETIN OF THE

arXiv:math/9207218v1 [math.CO] 1 Jul 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 27, Number 1, July 1992, Pages 148-153RATIONAL FUNCTION CERTIFICATIONOF MULTISUM/INTEGRAL/“q” IDENTITIESHerbert S. Wilf and Doron ZeilbergerAbstract. The method of rational function certification for proving terminating hy-pergeometric identities is extended from single sums or integrals to multi-integral/sumsand “q” integral/sums.1.

IntroductionSpecial functions have been defined by Richard Askey [As2] as “functions thatoccur often enough to merit a name,” while Tur´an [As1, p. 47] defined them as“useful functions.” The impact of these special functions on classical mathematicsand physics can be gauged by the stature of those whose names they bear: Bessel,Gauss, Hermite, Jacobi, Legendre, Tschebycheff, to name a few. It turns out thatmost special functions are of hypergeometric type, which is to say that they can bewritten as a sum in which the summand is a hypergeometric term.

Also of greatinterest are the so-called q-analogs of special functions and hypergeometric series,called q-series. These have many applications to number theory, combinatorics,physics, group theory [An], and other areas of science and mathematics.There are countless identities relating special functions (e.g., [PBM, R, An, As1]).In addition to their intrinsic interest, some of them imply important properties ofthese special functions, which in turn sometimes imply deep theorems elsewhere inmathematics (e.g., [deB, Ap]).

Just as important for mathematics are the extremelysuccessful attempts to instill meaning and insight, both representation-theoretic(e.g., [Mi]) and combinatorial (e.g., [Fo]), into these identities.The general theory of multivariate hypergeometric functions is currently a veryactive field, rooted in multivariate statistics and the physics of angular momentum.A very novel and fruitful approach is currently being pursued by Gelfand, Kapranov,Zelevinsky (e.g., [GKZ]) and their collaborators.We now know [Z1, WZ1] that terminating identities involving sums and integralsof products of special functions of hypergeometric type can be proved by a finitealgorithm, viz., find recurrence or differential equations that are satisfied by the leftand the right sides of the claimed identity (they always exist), and then compute1991 Mathematics Subject Classification. Primary 05A19, 05A10, 11B65, 33A30, 33A99; Sec-ondary 05A30, 33A65, 35N, 39A10.Key words and phrases.

Identities, hypergeometric, holonomic, recurrence relation, q-sums,multisums, constant term identities, Selberg and Mehta-Dyson integrals.Received by the editors December 2, 1991 and, in revised form, January 28, 1992The first author was supported in part by the U. S. Office of Naval ResearchThe second author was supported in part by the U. S. National Science Foundationc⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2H. S. WILF AND DORON ZEILBERGERenough initial values of the two sides to assure that the two recurrence or differ-ential equations have the same solution.

We know further that for one-variablehypergeometric sums [WZ1, WZ2, Z2, Z3] and single “hyperexponential” integrals[AZ] there are efficient algorithms for doing this, so computers really can do thejob.Here we announce fast, efficient algorithms for such identities that involve multi-ple sums and integrals of products of special functions of hypergeometric type. Wealso announce the algorithmic provability of single- and multivariate q-identitiesand, furthermore, that the algorithms are fast and efficient.

The proofs that aregenerated by these programs are extremely short, and human beings can verifythem easily.Further we show how to extend to “explicitly-evaluable” multiple sums and inte-grals the notions of rational function certification, WZ-pair, and companion identitythat were introduced in [WZ1] and [WZ2] for single sums.The present theory applies directly only to terminating identities, by which wemean that for any specific numerical assignment of the auxiliary parameters theintegral-sum is trivially evaluable. However, very often nonterminating identitiesare limiting cases or “analytic continuations” of terminating ones, so our resultsalso bear on them.Our method proves (or refutes) any such given conjectured identity, but humansare still needed to conjecture interesting ones.Full details and many examples will appear elsewhere [WZ3].

Our Maple pro-grams are available from .2. Hypergeometric multivariate identitiesFor a function F(k, y) of r discrete variables k and s continuous variables ylet Ki be the operator defined by KiF(k, y) = F(k1, .

. .

, ki + 1, . .

. , kr, y), and letDj = ∂/∂yj, for i = 1, .

. .

, r and j = 1, . .

. , s.Definition.

A function F(k, y) is a hypergeometric term if KiF/F (i = 1, . .

. , r)and DjF/F (j = 1, .

. .

, s) are all rational functions of (k, y).A sequence of special functions of hypergeometric type (in one variable) is asequence that is of the form Pn(x) := Pk F(n, k)xk where F is a hypergeometricterm.A typical identity in the theory of special functions involves multiple integral-sums of products of polynomials of hypergeometric type. After a full expansion,such an identity is of the form “left side”=“right side” where both sides are ofthe form PkRy F(k, n, x, y)dy.

Here F is a hypergeometric term in the discretemultivariables (k, n) and the continuous multivariables (x, y).It might happenthat one of the sides, say the right side, has no P’s orR’s in it, i.e., it is alreadyhypergeometric, in which case we speak of explicit evaluation.Finally, we also require that our integrand/summands be holonomic. This iscertainly the case for what we call proper-hypergeometric terms, which for purelydiscrete summands, look likeF(n, k) =Q(ain + bi · k + ci)!Q(uin + vi · k + wi)!P(n, k)ξk,(P-H)

MULTIVARIATE HYPERGEOMETRIC IDENTITIES3where the a’s and u’s are specific integers, b and v are vectors of specific integerentries, the c’s and the w’s are complex numbers that may depend upon additionalparameters, P is a polynomial in k, and ξ is a vector of parameters.This allows us to use the holonomic theory of [Z1]. However for pure multisums,we give proofs of our results, with effective bounds, that are self-contained andindependent of the theory of holonomic functions.3.

The fundamental theoremLet ∆i = Ki −1 be the forward difference operator in ki, and let N be theforward shift in n: Nf(n) = f(n + 1).Theorem 1. Let F(n, k, y) (resp.

F(x, k, y)) be a proper-hypergeometric term,or more generally, a holonomic hypergeometric term, in (k, y) and n (resp. x),where n, k are discrete and x, y are continuous variables.

Then there exist a lin-ear ordinary recurrence (resp. differential ) operator P(N, n) (resp.

P(Dx, x)) withpolynomial coefficients and rational functions R1, . .

. , Rr, S1, .

. .

, Ss such that(1)P(N, n)F(resp. P(Dx, x)F) =rXi=1∆ki(RiF) +sXj=1Dyj(SjF).For proper-hypergeometric F we give explicit a priori bounds for the order ofthe operator P. By summation-integration of (1) over k, y we obtain the followingCorollary A.

If F(n, k, y) (resp. F(x, k, y)) is as above and of compact supportin (k, y) for every fixed n (resp.

x) then(2)f(n) (resp. f(x)) :=XkZyFdysatisfies a linear recurrence (resp.

differential ) equation with polynomial coefficients(3)P(N, n)f(n) ≡0(resp. P(Dx, x)f(x) ≡0).The denominators of the rational functions R, S can be predicted for any givenF, and an upper bound for the order of the operator P can be given in advance.Hence by assuming the operator and the rational functions in the most generalform with those denominators and with that order, the determination of the un-known operator and rational functions quickly reduces to solving a system of linearequations with symbolic coefficients.

By [W] this reduces to solving such a systemwith numerical coefficients, and that in turn reduces to solving linear systems withinteger coefficients, a problem for which fast parallelizable algorithms exist [CC].Using the terminology of [WZ1, WZ2] one can say that the rational functions(R, S) certify the recurrence (resp. the differential equation) (3).Corollary B.

Any identity of the form “left side”=“right side,” where both sideshave the form (2) and F is a proper-hypergeometric or holonomic-hypergeometricterm, has a two-line elementary proof, constructible by a computer and verifiableby a human or a computer.

4H. S. WILF AND DORON ZEILBERGER4.

Sketch of the proof of the theoremFrom the general theory of [Z1] we can find a linear partial recurrence-differentialoperator T (n, N, K, Dy), independent of k, y, that annihilates F.For proper-hypergeometric functions we have a completely elementary proof that also givesexplicit upper bounds for the orders in N and K.This elementary proof wasobtained by extending to multisums-integrals Sister Celine Fasenmyer’s technique[Fa, R] as systematized by Verbaeten [V].Once the operator T has been found, we write(4)T (n, N, K, Dy) = P(N, n) +rXi=1(Ki −1)Ti(N, K, D) +sXj=1Dj bTj(N, K, D).It is easy to see that this is always possible. Next set(5)Gi(n, k, y) := −TiF(n, k, y)(i = 1, .

. .

, r),bGj(n, k, y) := −bTjF(n, k, y)(j = 1, . .

. , s).Since F is hypergeometric, the Gi and the bGj are rational multiples of F: Gi =RiF, bGj = SjF.

Now apply (4) to F(n, k, y), remembering that T F = 0, to get(1).□5. Discrete and continuous q-analoguesThe above extends to multivariate q-hypergeometric identities.

Let Qj be theoperator that acts on F by replacing yj by qyj wherever it appears.Then wesay that a function F(k, y) is a q-hypergeometric term if for each i = 1, . .

. , rand j = 1, .

. .

, s, it is true that KiF/F and QjF/F are rational functions of(q, qk1, . .

. , qkr, y1, .

. .

, ys). There is also a natural definition of q-proper-hypergeometric,which is given in [WZ3].The fundamental theorem still holds, where the integration is replaced either byJackson’s q-integration [An] or by an ordinary contour integral, or, in the case ofa formal Laurent series, by the action of taking “constant term of.” Macdonald’sq-constant term conjectures ([Ma], see [Gu, GG] for a recent update) for everyspecific root system, fall under the present heading.6.

Explicit closed-form identities:WZ-tuples and companion identitiesIn the case where the identity “left side”=“right side” is such that the right sidedoes not contain any ‘P’ or ‘R’ signs, i.e., is of closed form, one has an explicitidentity. If the right side is nonzero one can divide through by it to get an identityof the formXkZyF(n, k, y) dy = 1.Since the summand satisfies (1), the left side, call it L(n), satisfies some linearrecurrence P(N, n)L(n) = 0, by Corollary A.

Often the operator P turns out to bethe minimal order recurrence that is satisfied by the sequence that is identically 1,

MULTIVARIATE HYPERGEOMETRIC IDENTITIES5viz. (N −1)L(n) = 0.

If that happens then if we let Gi := −RiF and Hj := −SjFwe find that (1) becomes(6)∆nF +rXi=1∆iGi +sXj=1DyjHj = 0.We call (F, G, H) a WZ-tuple.It generalizes the idea of WZ-pair developed in[WZ1, WZ2]. Recall that a WZ pair gave, as a bonus, a new identity, the companionidentity.

Here, if we sum-integrate (6) w.r.t. all of the variables except one, weget a new identity for each choice of that one variable, for a total of r + s newcompanion identities altogether!7.

Example: The Hille-Hardy Bilinear Formulafor Laguerre PolynomialsAs an example we will now show the computer proof of the Hille-Hardy formula[R, Theorem 69, p. 212], namely,n! (α+1)n L(α)n (x)L(α)n (y) =12πiR|u|=ǫ u−n−1(1 −u)−α−1 expn−(x+y)u(1−u)o× Xmim!

(α + 1)mxyu(1 −u)2m!du.Many other examples appear in [WZ3].To this end, it is enough to prove that the right side is annihilated by the well-known second-order differential operator annihilating the Laguerre polynomials,both w.r.t. x and y.

Of course, by symmetry, it suffices to do it only for x, but thecomputer does not mind doing it for both x and y. We still need to prove that theinitial conditions match, but they are just the usual defining generating functionfor the Laguerre polynomials.

The computer output was as follows.Theorem. LetF(u, m, x) := (1 −u)−α−1 exp (−(x + y)u/(1 −u))(xyu/(1 −u)2)mun+1m!Γ(α + 1 + m),and let a(x) be its contour integral w.r.t.

u and sum w.r.t. m. Let Dx be differen-tiation w.r.t.

x. The function a(x) satisfies the differential equation(n + (α + 1 −x)Dx + xD2x)a(x) = 0.Proof.

It is routinely verifiable that(n + (α + 1 −x)Dx + xD2x)F(u, m, x)= Du(−uF(u, m, x)) + ∆m(−(m(α + m)/x)F(u, m, x))and the result follows by integrating w.r.t. u and summing w.r.t.

m.□Remark. The phrase “routinely verifiable” in the above means that after carryingout the indicated differentiation and differencing, and after dividing through by Fand clearing denominators, what will remain will be a trivially verifiable polynomialidentity.

6H. S. WILF AND DORON ZEILBERGERAcknowledgmentMany thanks are due to Dr.James C. T. Pool, head of Drexel University’sDepartment of Mathematics and Computer Science, for his kind and generous per-mission to use his department’s computer facilities.References[AZ]G. Almkvist and D. Zeilberger, The method of differentiating under the integral sign, J.Symbolic Comp.

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Soc.,Providence, RI, 1986.[Ap]R. Ap´ery, Irrationalit´e de ζ(3), Asterisque 61 (1979), 11–13.[As1]R.

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ICM,Warsaw, Aug. 16-24, 1983, Varsovie, 1983, pp. 1541–1553.[GG]F.

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24 (1991), 343–347.[GKZ]I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Generalized Euler integrals andA-hypergeometric functions, Adv.

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4,Addison-Wesley, London, 1977.[PBM]A. P. Prudnikov, Yu.

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EUROSAM’74, ACM-SIGSAM Bull. 8 (1974), 96–98.[W]H.

S. Wilf, Finding the kernel of a symbolic matrix, preprint.[WZ1]H. S. Wilf and D. Zeilberger, Rational functions certify combinatorial identities, J. Amer.Math.

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[WZ2], Towards computerized proofs of identities, Bull. Amer.

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23 (1990),77–84. [WZ3], An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/inte-gral identities, Inventiones Math.

(to appear).[Z1]D. Zeilberger, A holonomic systems approach to special functions identities, J. Comp.Appl.

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11 (1991), 195–204.University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395E-mail address: wilf@central.cis.upenn.eduTemple University, Philadelphia, Pennsylvania 19122-2585E-mail address: zeilberg@euclid.math.temple.edu

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